#### Geometry & Topology, Vol. 5 (2001)
Paper no. 18, pages 551--578.

## Homology surgery and invariants of 3-manifolds

### Stavros Garoufalidis, Jerome Levine

**Abstract**.
We introduce a homology surgery problem in dimension 3 which has the
property that the vanishing of its algebraic obstruction leads to a
canonical class of \pi-algebraically-split links in 3-manifolds with
fundamental group \pi . Using this class of links, we define a theory
of finite type invariants of 3-manifolds in such a way that invariants
of degree 0 are precisely those of conventional algebraic topology and
surgery theory. When finite type invariants are reformulated in terms
of clovers, we deduce upper bounds for the number of invariants in
terms of \pi-decorated trivalent graphs. We also consider an
associated notion of surgery equivalence of \pi-algebraically split
links and prove a classification theorem using a generalization of
Milnor's \mu-invariants to this class of links.
**Keywords**.
Homology surgery, finte type invariants, 3-manifolds, clovers

**AMS subject classification**.
Primary: 57N10.
Secondary: 57M25.

**DOI:** 10.2140/gt.2001.5.551

**E-print:** `arXiv:math.GT/0005280`

Submitted to GT on 31 May 2000.
(Revised 2 May 2001.)
Paper accepted 9 June 01.
Paper published 17 June 2001.

Notes on file formats
Stavros Garoufalidis, Jerome Levine

School of Mathematics

Georgia Institute of Technology

Atlanta, GA 30332-0160, USA

Department of Mathematics

Brandeis University

Waltham, MA 02254-9110, USA

Email: stavros@math.gatech.edu, levine@brandeis.edu

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